A less general reply:
Similar to Noveau the 3D feature set provided by the free Radeon driver is sometimes / often not good enough to run more demanding games (be it WINE or native). YMMV...depending on driver version, WINE version and the AMD card used.

The best place to get valuable informations for running games under WINE is often enough the Wine AppDB. Check out the entries for your game. May be someone with identical or similar hardware had the same issue and even has posted a solution.

For the case when open drivers are insufficient, I'd like to know a more detailed reason. like which feature is lacking?
For one thing, I'm curious, for another, It may help me to file a better bug report.

The first step is to know what opengl feature a game need.
But it is a big obstacle.
In games docs, there's generally only some simplest description like "requires directx 9.0c".

The second step is to know mesa's feature status.
It is easier (relatively), at least there's code and comment.

Below is how to do this in more detail in my imagination:

for games using opengl:
1. list all the opengl api a executable file calls by a static analyse. (which command?)
2. read mesa doc to see which mesa feature every gl-api maps to. (which doc?)
3. read mesa doc to see the feature status. (mesa feature status page)
4. read mesa doc to see if the gl-api is fully implemented (which doc?)

for games using directx:
1. list all the directx api a executable file calls by a static analyse. (which command?)
2. read wine doc to see what gl feature is used to implement a dx-api. (which doc?)
3. read mesa doc to see the gl feature status.

Generally, there's only error phenomenon and some test results on wine-appdb, no deeper analyze.
So I don't think wine-appdb is helpful on this level.